Finding min/max of quadratic bezier with CoreGraphics

Question

I am using CoreGraphics to draw a quadratic bezier but want to computer the min/max value of the curve. I am not from a mathematical background so this has become a bit troublesome. Does anyone have any articles or ideas about how to solve this?

Answer 1

For a quadratic Bezier, this is actually quite simple.

Define your three control points as P0 = (x0,y0), P1 = (x1,y1) and P2 = (x2,y2). To find the extrema in x, solve this equation:

t = (x0 - x1) / (x0 - 2*x1 + x2)

If 0 <= t <= 1, then evaluate your curve at t and store the location as Px. Do the same thing for y:

t = (y0 - y1) / (y0 - 2*y1 + y2)

Again, if 0 <= t <= 1, evaluate your curve at t and store the location as Py. Finally, find the axis-aligned bounding box containing P0, P2, Px (if found) and Py (if found). This bounding box will also tightly bound your 2D quadratic Bezier curve.

Answer 2

Calculus gives the standard box of tricks for finding the min/max of continuous, differentiable curves.

Here is a sample discussion.